Neutrosophic goal programming technique with bio inspired algorithms for crop land allocation problem

In agriculture, crop planning and land distribution have been important research subjects. The distribution of land involves several multi-functional tasks, such as maximizing output and profit and minimizing costs. These functions are influenced by a variety of uncertain elements, including yield, crop price, and indeterminate factors like seed growth and suitable fertilizer. In order to address this problem, other researchers have used fuzzy and intuitionistic fuzzy optimization approaches, which did not include the indeterminacy membership functions. However, the neutrosophic optimization technique addresses the problem by using individual truth, falsity, and indeterminacy membership functions. So, to improve the optimal solution, the Neutrosophic Goal Programming (NGP) problem with hexagonal intuitionistic parameters is employed in this study. The membership functions for truth, indeterminacy, and falsity are constructed using hyperbolic, exponential, and linear membership functions. Minimizing the under deviations of truth, over deviations of indeterminacy, and falsity yields the NGP achievement function, which is used to attain optimal expenditure, production, and profit under the constraints of labour, land, food requirements, and water. Bio-inspired computing has been a major research topic in recent years. Optimization is mostly accomplished through the use of bio-inspired algorithms, which draw inspiration from natural behaviour. Bio-inspired algorithms are highly efficient in exploring large solution spaces, and helps to manage trade-offs between various goals, and providing the global optimal solution. Consequently, bio-inspired algorithms such as Grey Wolf Optimization (GWO), Social Group Optimization (SGO), and Particle Swarm Optimization (PSO) are employed in the current work to determine the global optimal solutions for the NGP achievement function. The data for the study was collected from the medium-sized farmers in Ariyalur District, Tamil Nadu, India. To illustrate the uniqueness and application of the developed method, the optimal solutions of the suggested method are compared with Zimmermann, Angelov, and Torabi techniques. The proposed technique demonstrates that the bioinspired algorithms’ optimal solution to the neutrosophic goal is superior to the existing approaches.

nature-inspired optimization methodology known as the Harris Hawks Optimizer (HHO).Harris's hawks' cooperative nature and chasing technique serve as the primary source of inspiration for HHO.HHO is a relatively new method that has drawn the interest of numerous academics for its ability to tackle a variety of optimization issues, including single-and multi-objective, unconstrained, constrained, continuous, discrete, linear, nonlinear, and mixed-integer nonlinear problems [27][28][29][30] .Hang Su et al. 31 have suggested a novel high-performance optimization technique that relies on the rime generation process.This approach effectively addresses intricate optimization issues, including feature selection and image segmentation.Using the aforementioned metaheuristic techniques, globally optimal solutions are developed for a wide range of real-world problems.Therefore, in this study, bio-inspired algorithms such as PSO, GWO, and SGO are used to discover the global optimal solution for the achievement function created for the optimal crop land allocation problem.

Motivation and contribution
Numerous academics have employed intuitionistic and fuzzy optimization techniques, which essentially take truth and falsity membership functions into account, to address the crop land allocation problem.However, several unpredictability can occur in agriculture, such as crop development, seed growth, and fertilizer use.In crop planning problems, fuzzy and intuitionistic fuzzy sets capture all types of uncertainty, but a modern framework is needed to explain indeterminate comprehension.Thus, the neutrosophic approach is a superior optimization technique to identify the best solution for an ambiguous and indeterminate problem.All of these are the motivations behind creating a new NGP achievement function to maximize farmer profitability.The following are the primary goals and contributions of the current study: • To optimize the production, expenditure, and profit of a medium-sized farm owner in Ariyalur district under the constraints of food requirements, labour, water, and land.• Creating a new NGP problem using linear and non-linear membership functions and using PSO, GWO, and SGO for obtaining the global optimal solution for the NGP achievement function.
India is the most populous nation, and its daily food needs are growing.The Indian government has launched several new programmes in an effort to meet the nation's rising food demand.Crop planning is essential for agriculture to control the demand for food.An efficient strategy for allocating crop land can control seasonality, soil fertility, and variability in productivity.The suggested method helps manage the uncertainty and indeterminacy of crop planning and finds the most acceptable solution by fusing bio-inspired algorithms with a neutrosophic goal.The proposed technique can also be used to find the optimal solution for the numerous real-world MOOPs with indeterminacy and unclear aspiration levels.

Literature review
Optimizing return with minimal expenditure on restricted land is a primary goal in agriculture.To boost crop yield, Meselu Tegenie et al. 32 recommended linear programming-based crop land allocation for small farm holders.Mahak Bhatia et al. 33 employed a linear programming approach to determine the best crop combination for the large farm owner in Jaipur in order to increase productivity.Since most farm planning issues are multiobjective in nature and LP is a single-objective optimization technique, the goal programming (GP) technique, one of the popular tools for multi-objective decision analysis, is used to solve the problem of land allocation planning for the optimal production of a variety of crops.To increase farmers production and returns, Elizabeth Gosling et al. 34 evaluated agroforestry systems using a goal-programming approach.Animesh Biswas et al. 35 and Sharma et al. 36 proposed pre-emptive and weighted fuzzy goal programming techniques for land use planning to improve the yield of farmers.Srivastava et al. 37 proposed a two-level fuzzy goal programming technique for the land allocation problem in the canal command area.For the sustainable crop pattern, Ramtin Joolaie et al. 38 proposed a fuzzy goal programming method with a random sampling approach.For the production distribution with minimum transportation cost and delivery time Srikant Gupta et al. 39 proposed efficient fuzzy goal programming which is the combination of goal program, fuzzy program and interactive program.Demmelash Mollalign Moges et al. 40 proposed a two-phase weighted intuitionistic fuzzy goal programming technique to solve an intuitionistic fuzzy multi-objective linear fractional optimization problem, and for the efficiency of the algorithm, the author applied it to an agricultural production planning problem.Angammal et al. 41 compared fuzzy and intuitionistic fuzzy optimization methods for three-season's crop land allocation problem.Bharati et al. 42 developed intuitionistic fuzzy optimization using fractional programming to maximize small farm holders profits and production.Intuitionistic fuzzy optimization was developed by Pawar et al. 43 to maximize net profit and employment generation with minimal cultivation cost.Real-life complications can lead to indecision or neutral thinking when making the best decisions.The degree of indeterminacy is also important in decision-making, along with membership and non-membership.Thus, Sajida Kousar et al. 44 offered neutrosophic optimization to maximize rice and wheat productivity and profit in Pakistan's kharif and rabi seasons.To minimize production and inventory holding costs and maximize net profit for the hardware firm's multi-product manufacturing problem, Mohammad Faizal Khan et al. 45 developed neutrosophic and intuitionistic fuzzy optimization.Angammal et al. 46 proposed an INPA with exponential and linear membership functions to optimize land allocation for medium-sized farm owners.Indrani Maiti et al. 47 proposed goal programming with interval parameters for the multi-level, multi-objective linear programming problem with the neutrosophic number parameter.Firoz Ahmad et al. 48addressed the multi-level, multi-objective fractional programming problem with a rough interval parameter using neutrosophic goal programming.A unique neutrosophic programming approach was proposed by Sapan Kumar Das et al. 49 to solve the linear programming problem with pentagonal neutrosophic number parameter.Suizhi Luo et al. 50 www.nature.com/scientificreports/interactive programming approaches to solve the multi-level programming problem in the neutrosophic environment, including the Score function based interactive approach, the TOPSIS based interactive approach, and the Multi Objective Optimization on the basis of Ratio Analysis (MOORA) based interactive approach.Firoz Ahmad et al. 51 proposed an interactive neutrosophic programming method to manage energy, food, and water security.Complex optimization problems can be solved with high-quality solutions using bio-inspired optimization methods.Therefore, Xiaoping Liu et al. 52 presented hybrid PSO, a new particle adjustment rule with genetic reproduction mechanisms, crossover and mutation with system dynamics for land distribution in Panyu, Guangdong, China.A modified social group optimization technique was presented by Swagato Das et al. 53 for damage assessments of modelled civil engineering structures.To maximize net benefits and minimize fertility usage Crow Search Algorithm (CSA) and PSO were combined by Sonal Jain et al. 54 Ashutosh Rath et al. 55 used swarm intelligence techniques including GA, cuckoo search (CS), and PSO to create a cropping pattern that maximized net return for the Hirakud command region in India.Zhidong Wang et al. 56 used GWO to find the pareto optimal solution for agricultural water allocation in Aksu Valley, Xinjiang, China.To predict the accurate fertilizer application ratio and improve the yield Cengcheng Chen et al. 57 proposed the novel multi strategy GWO algorithm.Alireza Donyaii et al. 58 proposed the multi-objective GWO algorithm to minimize vulnerability and maximize the reliability of the optimum rules for operating Golestan Dam, located in Iran, under climate change conditions.From the above literature review, it is clear that various agricultural problems have been solved by using LP, GP, fuzzy, intuitionistic fuzzy, neutrosophic approach, and bioinspired algorithms.But for crop land allocation problems, which involve many uncertainty and indeterminacy factors, the NGP with linear and nonlinear membership functions is not yet used.Thus, for determining the most suitable land allocation for each crop, the neutrosophic goal programming approach is offered here, and also PSO, SGO, and GWO employed in this study to get the global optimal solution for the achievement function.

Study area
The district of Ariyalur, which has excellent agricultural practices, provided all of the agricultural data needed for this study.The Ariyalur District covers a land area of 1,93,338 hectares.1,11,874 hectares are under cultivation.
There are approximately 66,738 hectares of rain-fed land and 45,136 hectares of irrigated land.The majority of inhabitants in this district work mostly in agriculture, where they cultivate a wide range of crops.1% of farmers in the district of Ariyalur are medium-sized farmers.The Ariyalur district agricultural land type, irrigation systems, and major cultivated crops are presented in Fig. 1.The medium farmer in Narasingapuram village, Ariyalur district, Tamil Nadu, provided the cultivation data for the present study.In that district, the 15-acre farmer cultivated paddy, cotton, and groundnut in the kharif; paddy, sweet corn, groundnut, and pearl millet

Methodology
Among all the methods discussed in the literature review, no one has developed a neutrosophic goal programming technique for crop land allocation problems that includes linear, hyperbolic, and exponential membership functions.Instead, many researchers employed intuitionistic fuzzy and fuzzy goal programming with linear membership functions and linear goal programming strategies for multi-objective crop land allocation problems.As a result, this paper's methodology attempts to address this problem.The problem of allocating agricultural land has intuitionistic parameters.Therefore, the author begins by estimating the intuitionistic parameter's value.After evaluation, the crisp neutrosophic objective function is produced by minimizing the over deviation of untruth and indeterminacy and the under deviation of truth.Bio-inspired optimization algorithms use nature's biological evolution to create robust solutions.Therefore, the crisp neutrosophic goal programming problem is solved by using some bio-inspired algorithms named PSO, GWO, and SGO.Although these three algorithms are frequently employed in real-world scenarios, they have not been utilised in the context of the neutrosophic goal programming technique to address the crop land allocation problem.The proposed methodology's process is depicted in Fig. 2.

Particle Swarm Optimization (PSO)
In 1995, James Kennedy and Russell C. Eberhart 59 created PSO, a stochastic optimization method based on swarm intelligence.PSO solves problems through social interaction.A swarm of particles searches the search space for the optimal solution.Each particle's velocity is dynamically changed by its movement experience and that of its neighbours.The global best (gbest pi ) is the particle with the highest fitness value, while the personal best (pbest pi ) is the particle's best position.The PSO algorithm optimizes by following (pbest pi ) and (gbest pi ) .All particles are updated at each step by the following equations.
where t is the number of iterations, p is the particle number, and i denotes the component number.y pi and v pi are particles' positions and velocities respectively, and w is the inertia weight, which controls the effect of previous velocities on the present velocity, c 1 and c 2 are two positive constants, named cognitive learning rate and social www.nature.com/scientificreports/learning rate, and r 1 and r 2 are uniformly distributed random numbers between 0 and 1.The pth particle's local best solution is pbest pi , while gbest pi represents the global best.Once the best position of all particles are improved after a significant number of generations, the PSO algorithm terminates.

Grey Wolf Optimization (GWO)
In 2014, Mirijalili et al. 60 presented GWO, a population-based meta heuristic algorithm that simulates grey wolf leadership and hunting.The top predators in the food chain are grey wolves.It lives in packs of 5-12 members.
The first social hierarchy level is alpha ( α ).The alpha wolf makes judgements about hunting, employment, sleeping, and more.Pack members must follow the alpha wolf 's directions.Beta ( β ) is the second level.Alpha wolves make decisions with beta wolves' aid.The beta supports the alpha's orders and provides feedback.Delta ( δ ) wolves are subordinates to the alpha and beta wolves but dominate the omega ( ω ) wolves, the lowest level in the pack's social order.Omega wolves are the pack's scapegoat and must submit to all superior wolves.Alpha, beta, and delta are the first, second, and third fittest solutions in grey wolf optimization (GWO).Omega represents the remaining solutions.Grey wolves surround prey during hunting, this encircling behaviour is expressed mathematically in the following equation.
where A and C are coefficient vectors, t is the current iteration, y P is the position vector of the prey, and y indicates the grey wolf 's position vector.A & C vectors are calculated as follows: where r 1 , r 2 are random vectors in [0,1] and a ′ s components drop linearly from 2 to 0. We assumed the alpha, beta, and delta have higher prey-locating knowledge in the grey wolf mathematical model.The first three best solutions are stored, and the other agents must update their positions based on the best search agent's position as stated in the following equations.
When prey stops moving, the grey wolf attacks.A is a random vector in [-2a, 2a].When |A| < 1 , the wolves exploit the prey, but when |A| > 1 , they must diverge to find a better prey.

Social Group Optimization (SGO)
SGO is a new optimization method that relies on humans' group-solving abilities, introduced by Satapathy and Naik 61 in 2016.In this approach, a group of people are selected and enhanced with knowledge of diverse capacities to solve a function.SGO contains two phases, the Improving and Acquiring phase that helps to find optimal solution for numerous real-life problems.

Improving phase
In this phase, the best person in the group raises everyone's knowledge level and is represented as follows: where C is the self-introspection parameter and C ∈ [0 1] , r is the random number between 0 and 1, y new , y old & y best are the group's new, old, and best individual's positions respectively.

Acquiring phase
In this phase, each group member learns from the best and other group members.The new solution is generated with the help of partner's position y p .The present study deals with the minimization problem, and the math- ematical expression is described as follows: where r 1 , r 2 ∈ [0 1] , which improves social behaviour by promoting the algorithm's randomness.In this scenario, y old is the individuals' improvement phase position.If y new improves objective function fitness, then it is accepted.

Neutrosophic approach in MOOP
The complexity of real life often leads to undecidability when making better decisions.The degree of indeterminacy plays a significant role in the decision-making process, in addition to the acceptance and rejection degrees.
As such, the feasible solution set includes the indeterminacy degree.NS is unique among uncertain decision sets since it has an independent degree of indeterminacy, which sets it apart from fuzzy and intuitionistic fuzzy y new = C.y old + r(y best − y old ) Vol.:(0123456789) www.nature.com/scientificreports/sets.A neutrosophic decision set (G N ) for the MOOP is formed by the intersection of integrated neutrosophic objects (O i ) and neutrosophic constraints (C j ) .It is explained as follows: where, T G N (y), I G N (y), F G N (y) represent the membership functions of truth, indeterminacy and falsity of deci- sion set G N .By minimizing the underdeviation of truth and the overdeviation of falsity and indeterminacy, the following proposed NGP is utilized to produce an achievement function for the MOOP.

Proposed neutrosophic goal programming technique (NGP)
In approaches to neutrosophic goal programming, each marginal evaluation is converted into a neutrosophic membership goals in accordance with their highest accomplishment levels.The truth membership function can be attained up to a maximum of unity.The highest degree of accomplishment for the indeterminacy membership function is half.The utmost achievement degree for the falsehood membership function is zero.To create a Neutrosophic membership goals, the lower and upper limit of the truth, indeterminacy, and falsity membership functions are created as follows: Maximization Minimization where U acc , L acc are the upper and lower acceptance of the truth membership function, y is the decision vari- able, (U are the upper and lower limits of truth, indeterminacy and falsity membership functions and α K & β K are the scaling factors, as well as α K , β K ∈ [0 1].
In multi objective goal programming problem (MOGPP), each objective function's marginal evaluation is represented by its corresponding membership function.Since a linear type membership function has a simpler form and more understandable implications, it is generally the most comprehensive and commonly used type.The linear-type membership function takes the constant marginal rate of satisfaction or dissatisfaction degrees towards an objective into account.However, a non-linear membership function may be used to reflect each target's desire level.The non-linear membership functions' adaptable nature makes it possible to calculate the marginal assessment of the degree of objective satisfaction.It also depends on the values of a few parameters, which are sufficient to carry out the decision-maker's plan effectively.Therefore, this study represents a fresh attempt to investigate the MOGPP in a neutrosophic setting with both linear and non-linear membership functions.The present study examines Linear, Hyperbolic, and Exponential membership functions.The truth, indeterminacy, and falsity membership functions of these three types are ( ) respectively, and they are described in the following sub sections.

Hyperbolic membership function
Vol.:(0123456789) For all objective functions, U i = L i .If U i = L i , in any circumstance then the membership value will be assumed to be 1.
In order to highest accomplishment level of truth, indeterminacy and falsity, the membership functions mentioned in Eqs.(1) to (6) Other researchers produced the achievement function for the goal programming problem by minimizing the over deviation of the falsity membership functions and the under deviations of the truth and indeterminacy group.However, the decision-maker in this study minimized the over achievement of the indeterminacy and falsity membership functions and the under achievement of truth to generate the achievement function.As a result, the neutrosophic goal programming problem's achievement function is expressed as follows: are the weights associated with the deviations of the membership goals.After the multi-objective optimization problem has been transformed into a single objective neutrosophic goal, bio-inspired algorithms GWO, PSO, and SGO are utilized to discover the best solution to the crisp neutrosophic goal programming problem.The crop land allocation problem mentioned in the study area is solved using the aforesaid methodology.

Computational algorithm
It is possible to formulate a real-world scenario into an optimization model using a computational process for any mathematical model.Using the neutrosophic goal programming technique and a bio-inspired algorithm, the following steps are taken to solve the linear multi-objective crop land allocation problem.
Step 1: Consider the multi-objective linear programming problem (MLPP) with 'r' objectives, 'm' constraints, and 'n' decision variables as follows: Step 2: Utilizing the provided constraints, determine each objective function's solution one at a time.Create a pay-off matrix using the optimum solution's information as follows: where (y 1 ), (y 2 ), . . .(y r ) are the solution sets of the objectives O 1 , O 2 , . . .O r respectively.
Step 3: Obtain the lower and upper bound of each objective functions and determine the lower and upper bound of truth, indeterminacy & falsity membership function.
Step 4: Create Linear, Exponential and Hyperbolic membership function by using the bounds mentioned in step 3.
Step 5: As stated in Eq. ( 7), construct a neutrosophic achievement function and goals using the membership functions generated in Step 4.
Step 6: Use bio-inspired algorithms such as GWO, PSO, SGO to find the optimal solution of the goal mentioned in step 5.

Results and discussion
In this study, the parameters of production, profit, expenditure, labour, and water requirements of the problem explained in the study area are regarded as hexagonal intuitionistic fuzzy numbers due to uncertainty and indeterminacy.The hexagonal intutionistic fuzzy numbers for expenditure, production, profit and labour per acre, water, and labour availability per season are shown from Table 1, 2, 3 and 4. The estimated value of the hexagonal intuitionistic fuzzy numbers are derived from the Thamaraiselvi approach 62 .Once the hexagonal intutionistic fuzzy numbers are converted into the estimated value, the objectives (production, profit, labour cost, seed cost, fertilizer and pesticides cost, and miscellaneous costs) and the constraints (land, water, labour, and food requirements) are modified as follows:  ) respectively.These positive and negative ideal solutions are used to form the membership functions for the present study.Following the generation of linear and non-linear membership functions, the NGP achievement function is produced.By using a neutrosophic goal program mentioned in Eq. ( 7), with linear membership functions mentioned in Eqs.(1) and ( 2) the above Eq.( 8) is converted into the following Eq.( 9) Using the neutrosophic goal program mentioned in Eq. ( 7), with exponential membership functions mentioned in Eqs. ( 3) and (4), Eq.( 8) is converted into the following Eq.( 10) All the constraints of Eq. (8).7), hyperbolic membership functions mentioned in Eqs. ( 5) and ( 6), Eq. ( 8) is converted into the following Eq.( 11) Using linear and non-linear membership functions as mentioned in Eqs.(1) to ( 6), the multi-objective optimization problem, which involves uncertainty and indeterminacy, is converted into the crisp single-objective neutrosophic optimization problem as mentioned in Eqs.(9) to (11).The objective functions of these three equations are called the achievement function of the problem.The main goal of this problem is to obtain the global optimal solution for these achievement functions.As mentioned earlier, bioinspired algorithms effectively search the search space and obtain the global optimal solution.Therefore, some of the bioinspired algorithms, namely PSO, GWO, and SGO, as explained under the methodology sections, are used to find the global optimal solution for the current problem.In this study, the number of populations, iterations, and runs for PSO, GWO, and SGO are fixed as the same, and they are (1000, 500, 20).Also, the acceleration constant (C1, C2) and the inertia weight w are fixed as (2, 2) and 1 in PSO, and the self-introspection parameter C is fixed as 0.25 in SGO.
The optimal solutions of the achievement functions are obtained using these algorithms, and they converge to zero as the solution approaches 500 iterations in 20 runs.The convergence graphs for the three cases are shown in Figs. 3, 4 and 5. From these convergence graphs, it is concluded that PSO, GWO, and SGO provide the best compromise solution to the achievement functions.Therefore, it will provide the most satisfactory solution for the objective functions of production, profit, and expenditure.Also, the computation time of these three algorithms used to obtain the global optimal solution for the achievement functions of NGPs' was almost the same.Thus, the best optimal solutions are classified by using the comparative analysis of the objective values, which is explained in the following section.For comparison, the optimal solutions of the objective functions are tabulated in Table 5. and the net profit obtained by the proposed and existing approaches and the farmer's original net profit are tabulated in Table 6.

Comparative analysis
In this analysis, the performances of PSO, GWO, and SGO have been compared for the crop planning problem with the same population size and iterations (1000 and 500, respectively).The optimal (production, profit, labour cost, seed cost, fertilizer and pesticides, miscellaneous cost, and net profit) obtained by using PSO for the NGP, which was created by using linear, exponential, and hyperbolic membership functions, are (145926. ) All the constraints of Eq. (8).
the net profit, PSO has provided a uniform and highly satisfactory solution to the NGP developed using linear and exponential membership functions.This is better than the net profit of the NGP generated using the hyperbolic membership function.Similarly, the optimal objective function values and net profit obtained using GWO and SGO for NGP generated using linear, exponential and hyperbolic membership functions are {(145146.} respectively.Comparing the net profit of NGP found using GWO, the net profit of NGP modeled with linear membership functions is higher than the net profit of NGP modeled with exponential and hyperbolic membership functions.Similarly, comparing the net profit of NGP found using SGO, the net profit of NGP modeled with exponential membership functions is higher than the net profit of NGP modeled with linear and hyperbolic membership functions.In the overall comparison, it is concluded that PSO provided a better net profit than GWO and SGO to the NGP.Also, for validation and superiority of the proposed method, its' findings are compared with those of Zimmermann 63 , Angelov 64 , and Torabi 65 .The optimal net profit obtained from these three approaches is the same and is Rs.3356462, and it is also compared with the farmer's original net profit of Rs. 2877399.These comparisons demonstrate that the profit obtained by the proposed method is more abundant and acceptable than the profit obtained by the existing approaches and the actual profit of the farmers.The comparative analysis of four types of expenditure and production and the comparative analysis of profit and net profit are shown in Figs. 6 and 7.

Conclusion
Numerous researchers have placed a great deal of emphasis on the crop planning problem due to its numerous risks and uncertainties.Without accounting for the indeterminacy component, a number of approaches have been devised to solve the land allocation problem.But in the current study, the achievement functions, which are generated by using a neutrosophic goal with independent linear, hyperbolic, and exponential truth, falsity, and indeterminacy membership functions, were solved in order to ascertain the optimal land allocation for the medium-sized farm-holders in the Ariyalur district.Additionally, PSO, GWO, and SGO were used to solve these achievement functions, and the optimal outputs are shown in Tables 5 and 6.Also, the comparative analysis of production and expenditure, profit, and net profit is shown in Figs. 6 and 7. From these tables and graphs, it is clear that the PSO algorithm provided a better compromise solution than the GWO and SGO for the NGP with a linear, hyperbolic, and exponential membership function.Also, the Zimmermann, Angelov, and Torabi and Hassini techniques were used to demonstrate the superiority of the proposed neutrosophic goal programming strategy.Comparing the results, the proposed strategy outperforms existing methods and the farmer's original output.The proposed method optimizes medium-sized farm-holder land allocation.However, this technology can also be used to improve state and national irrigation systems, solve crop combination problems, and find 14:21565 | https://doi.org/10.1038/s41598-024-69487-0

Figure 2 .
Figure 2. Flowchart of neutrosophic goal with bio inspired algorithm.

Figure 3 .
Figure 3. PSO Convergence of fitness value using linear, exponential and hyperbolic membership function.

Figure 4 .
Figure 4. GWO Convergence of fitness value using linear, exponential and hyperbolic membership function.

Figure 5 .
Figure 5. SGO Convergence of fitness value using linear, exponential and hyperbolic membership function.

Figure 6 .
Figure 6.Comparative analysis of production and expenditure.

Figure 7 .
Figure 7. Comparative analysis of profit and net profit.
www.nature.com/scientificreports/ in the rabi; and paddy, brinjal, sesame, and onion in the summer season.This research optimizes productivity, profit, and other costs, including labour, seed, fertilizer, pesticides, and miscellaneous charges within land, water, labour, and food restrictions.The methods utilized to identify the best solution and to produce the neutrosophic achievement function will be covered in the following section.

Table 6 .
Net profit of farmer, proposed and existing approach.